Average Error: 7.9 → 0.1

Time: 2.2s

Precision: binary64

Cost: 448

\[\frac{x \cdot y}{y + 1} \]

↓

\[\frac{x}{1 + \frac{1}{y}} \]

(FPCore (x y) :precision binary64 (/ (* x y) (+ y 1.0)))

↓

(FPCore (x y) :precision binary64 (/ x (+ 1.0 (/ 1.0 y))))

double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

double code(double x, double y) {
	return x / (1.0 + (1.0 / y));
}

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (y + 1.0d0)
end function

↓

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x / (1.0d0 + (1.0d0 / y))
end function

public static double code(double x, double y) {
	return (x * y) / (y + 1.0);
}

↓

public static double code(double x, double y) {
	return x / (1.0 + (1.0 / y));
}

def code(x, y):
	return (x * y) / (y + 1.0)

↓

def code(x, y):
	return x / (1.0 + (1.0 / y))

function code(x, y)
	return Float64(Float64(x * y) / Float64(y + 1.0))
end

↓

function code(x, y)
	return Float64(x / Float64(1.0 + Float64(1.0 / y)))
end

function tmp = code(x, y)
	tmp = (x * y) / (y + 1.0);
end

↓

function tmp = code(x, y)
	tmp = x / (1.0 + (1.0 / y));
end

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_] := N[(x / N[(1.0 + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{x \cdot y}{y + 1}

↓

\frac{x}{1 + \frac{1}{y}}

Try it out

In
Out

Enter valid numbers for all inputs

Target

Original	7.9
Target	0.0
Herbie	0.1

\[\begin{array}{l} \mathbf{if}\;y < -3693.8482788297247:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \mathbf{elif}\;y < 6799310503.41891:\\ \;\;\;\;\frac{x \cdot y}{y + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y} - \left(\frac{x}{y} - x\right)\\ \end{array} \]

Derivation

Initial program 7.9
\[\frac{x \cdot y}{y + 1} \]
Taylor expanded in x around 0 7.9
\[\leadsto \color{blue}{\frac{y \cdot x}{1 + y}} \]
Simplified0.1
\[\leadsto \color{blue}{\frac{x}{1 - \frac{-1}{y}}} \]
Proof
(/.f64 x (-.f64 1 (/.f64 -1 y))): 0 points increase in error, 0 points decrease in error
(/.f64 x (-.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) (/.f64 -1 y))): 0 points increase in error, 0 points decrease in error
(/.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y -1) y))): 2 points increase in error, 0 points decrease in error
(/.f64 x (/.f64 (Rewrite=> sub-neg_binary64 (+.f64 y (neg.f64 -1))) y)): 0 points increase in error, 0 points decrease in error
(/.f64 x (/.f64 (+.f64 y (Rewrite=> metadata-eval 1)) y)): 0 points increase in error, 0 points decrease in error
(Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x y) (+.f64 y 1))): 48 points increase in error, 25 points decrease in error
(/.f64 (Rewrite=> *-commutative_binary64 (*.f64 y x)) (+.f64 y 1)): 0 points increase in error, 0 points decrease in error
(/.f64 (*.f64 y x) (Rewrite<= +-commutative_binary64 (+.f64 1 y))): 0 points increase in error, 0 points decrease in error
Final simplification0.1
\[\leadsto \frac{x}{1 + \frac{1}{y}} \]

Alternatives

Alternative 1
Error	1.0
Cost	584

\[\begin{array}{l} t_0 := x - \frac{x}{y}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;x \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	30.8
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022337 
(FPCore (x y)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, B"
  :precision binary64

  :herbie-target
  (if (< y -3693.8482788297247) (- (/ x (* y y)) (- (/ x y) x)) (if (< y 6799310503.41891) (/ (* x y) (+ y 1.0)) (- (/ x (* y y)) (- (/ x y) x))))

  (/ (* x y) (+ y 1.0)))